I chucked out some organic household waste into the garden this winter - then overturned the garden this spring and planted flowers.
Turns out a potato that I’d thrown in November spouted and grew through the compost this year.
I would like to harvest this single potato plant - but I don;t know when the seaon is correct.
So - when should I look to dig up the potatoes? I figure it’s before the plant dies, yes?
Nope, you can leave potatoes in the ground after the plant dies, and they will actually keep longer. At least until the following spring, when they will sprout into new potato plants.
New potatoes/earlies are usually harvested when the plant produces flowers (early to mid summer).
Maincrop potatoes are generally harvested as the plant begins to turn yellow and wither.
Thanks for that. 
We don’t have much room to grow our own, so it’s interesting that we can get something from the garden to eat.
I have been growing potatoes in garbage cans for about 10 years now. Had a nice baked potato for dinner last night and it was harvested just minutes prior to baking. Here is a site with info for getting started. My potato cans have doors located low, I can harvest the bottom spuds without affecting the upper ones. I grow 3 plants per can and this year I am growing 7 different kinds of potatoes. This years harvest should total about 40 pounds of spuds, we won’t have to buy any till some time next year. The only problem I ever had was when a couple of raccoons dug up the potatoes when the cans were only half full. This happened a few years ago but they haven’t been back since.
I’m pretty sure that potatoes don’t “ripen” as far as eating them is concerned. There might be a time in the development of the tuber that the “eyes” or seeds are not fully developed, but the tuber itself isn’t sour, or whatever, like citrus fruits, apples, etc.
And, as others have said, you don’t have to dig them up right away either.
But this one is growing in the compost heap; the tubers are already at risk from slugs and other beasties in there, it is quite unlikely that the tubers would survive right through the winter without being eaten.
I grow potatoes in a plastic dustbin on my patio - it is good this way because I start with it only about 1/3 full and add more compost as the plants grow - ‘earthing up’ as I go. I use ordinary container compost (not peat-based though) and it has the added advantage of producing very clean and regular-shaped tubers, ideal for showing.